The results from the previous section point out the fact that binary labeling strongly affects the BICM-GMI. Moreover, we have seen that the BICM-GMI curves for a given binary labeling intersect each other for different constellation sizes, which naturally leads to a quite general question: What is the best constellation, binary labeling, and input distribution for the BICM at a given SNR? Once this question is answered, approaching the fundamental limit will depend only on a good design of the binary encoder/decoder.

To formalize the previous question, and in analogy with (4.25), we define the *BICM pseudo-capacity* for a given constellation size as

where the optimization is over the constellation and its binary labeling, as well as over the vector of bitwise probabilities that induce a symbol-wise PMF (via (2.72)) satisfying the energy constraint . We use the name BICM pseudo-capacity because, in theory, the quantity in (4.64) might not be the *largest achievable rate*, which simply comes from the fact that the BICM-GMI is not the largest achievable rate ...

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